Question

Calculate the inductance of a unit length of a double tape line as shown in figure. The tapes are separated by a distance $\mathrm{h}$ which is considerably less than their width $\mathrm{b}$.

Answer 1

Step 1: Given data

Distance = $\mathrm{h}$

Width = $\mathrm{b}$

Step 2: Concept used

By using Maxwell's ampere law, $\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{b}}$

where,

$\mathrm{B}$ is the magnetic field.

${\mathrm{\mu }}_0$ is the permeability of free space$(4\mathrm{\pi }\times {10}^{-7}\mathrm{N}/{\mathrm{A}}^2)$.

$\mathrm{I}$ is the current.

$\mathrm{b}$ is the width of the given double tape.

$\mathrm{\varphi }=2\mathrm{BA}$

Step 3: Calculation of inductance of unit length

$\text{∮}\overrightarrow{\mathrm{B}}\text{×}\overrightarrow{\mathrm{dl}}={\mathrm{\mu }}_0\mathrm{I}$

$\Rightarrow \mathrm{B}\times 2\mathrm{b}={\mathrm{\mu }}_0\mathrm{I}$

$\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{b}}$

Step 4: Find the magnetic flux

Since, net field = $2\mathrm{B}$ Magnetic flux passing through this double tape

$\mathrm{\varphi }=2\mathrm{BA}$

$=2\mathrm{B}\left(1\mathrm{h}\right)$

$\mathrm{\varphi }=\frac{{\mathrm{\mu }}_0\mathrm{I}}{\mathrm{b}}1\mathrm{h}$

Then, inductance$\left(\mathrm{L}\right)=\frac{\mathrm{\varphi }}{\mathrm{I}}$

$=\frac{{\mathrm{\mu }}_{0}1\mathrm{h}}{\mathrm{b}}$

$=\frac{\mathrm{L}}{1}$

$=\frac{{\mathrm{\mu }}_0\mathrm{h}}{\mathrm{B}}$

Therefore, the inductance of unit length is $\frac{{\mathrm{\mu }}_0\mathrm{h}}{\mathrm{B}}$.